Variational lower bounds estimators can be categorized in generative and discriminative approaches. The former attempts to separately estimate the joint and marginal distributions, \(p_{XY}(\mathbf {x},\mathbf {y})\)  and \(p_{X}(\mathbf {x})\cdot p_{Y}(\mathbf {y})\) , respectively. In [1]}, a variational distribution \(q_{X,\phi }(\mathbf {x}|\mathbf {y})\)  is introduced to replace the intractable conditional one \(p_{X}(\mathbf {x}|\mathbf {y})\) , leading to the bound
\(I(X;Y) \ge & I_{BA}(X;Y) \nonumber \\= & \mathbb {E}_{(\mathbf {x},\mathbf {y})\sim p_{XY}(\mathbf {x},\mathbf {y})}\bigl [\log ( q_{X,\phi }(\mathbf {x}|\mathbf {y}))- \log ( p_{X}(\mathbf {x})) \bigr ] ,\) 
